A Finite Difference Scheme for Blow-Up Solutions of Nonlinear Wave Equations

<正>We consider a finite difference scheme for a nonlinear wave equation,whose solutions may lose their smoothness in finite time,i.e.,blow up in finite time.In order to numerically reproduce blow-up solutions,we propose a rule for a time-stepping, which is a variant of what was successfully used in the case of nonlinear parabolic equations.A numerical blow-up time is defined and is proved to converge,under a certain hypothesis,to the real blow-up time as the grid size tends to zero.     

KAM Theory for Partial Differential Equations

In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.     

傅种孙先生在北京师大附中

1 傅种孙先生简介 1.1 傅先生基本情况介绍 傅种孙先生(1898～1962)1898年2月27日生于江西省高安县.1920年在北京高等师范学校数理部毕业,留母校附中(即现北京师大附中)任教;1921年任母校数理部的讲师;1928年任教授:抗日战争以前,曾先后兼任北京女子师范大学,北平大学女子文理学院,北京大学,辅仁大学教授;1933年当选北平(北京)市数学会理事长秘书;1935年当选中国数学会评议委员兼<数学杂志>编辑;1945年11月至1946年8月在牛津大学、1946年9月至1947年11月在剑桥大学考察;1947～1962年任北京师范大学数学系教授,1956年前曾兼系主任;在1949～1957年间,曾任北京师范大学教务长(3年)及副校长(5年);1952～1957年任北京市人民代表大会代表,中国数学会及其北京市分会常务理事,<中国数学杂志>及其后身<数学通报>总编辑;1962年1月18日病逝于北京.     

Local Unstable Entropy and Local Unstable Pressure for Partially Hyperbolic Endomorphisms*

In this paper, local unstable metric entropy, local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. Specially, two variational principles concerning relationships among the above mentioned numbers are formulated.     

Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by Nonsmooth Elliptic Equations

In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.     

A Direct Method of Moving Planes to Fractional Power SubLaplace Equations on the Heisenberg Group

We give the direct method of moving planes for solutions to the conformally invariant fractional power sub Laplace equation on the Heisenberg group.The method is based on four maximum principles derived here.Then symmetry and nonexistence of positive cylindrical solutions are proved.     

智慧窗

1 趣解等式七个文字与三个拼音字头组成了一道加法等式,你能否把它们分别换成0～9的数字. (相同文字、字母必须要换相同的数字)使其等式成立吗? (上海市长宁路476弄8号1602室(200042) 张刘福) 2 质数趣题请将5～20各数分别填入圆圈内,使每相邻的两个数之和都是质数.试试看,你能填出     

Hölder,Sobolev,weak-type and BMO estimates in mixed-norm with weights for parabolic equations

We prove weighted mixed-norm Lqt(W2,px)and Lqt(C2,αx)estimates for 10,x∈Rn.x∈Rn,The coefficients a(t)=(aij(t))are just bounded,measurable,symmetric and uniformly elliptic.Furthermore,we show strong,weak type and BMO-Sobolev estimates with parabolic Muckenhoupt weights.It is quite remarkable that most of our results are new even for the classical heat equation?tu?Δu+u=f.     

ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OFΓANDζ

In this paper,we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function.It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.     

Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs

A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex.A proper total-k-coloring of G is called neighbor sum distinguishing if∑_c(u)≠∑_c(v)for each edge uv∈E(G),where∑_c(v)denote the sum of the color of a vertex v and the colors of edges incident with v.The least number k needed for such a total coloring of G,denoted byχ∑"is the neighbor sum distinguishing total chromatic number.Pilsniak and Wozniak conjecturedχ∑"(G)≤Δ(G)+3 for any simple graph with maximum degreeΔ(G).By using the famous Combinatorial Nullstellensatz,we prove that above conjecture holds for any triangle free IC-planar graph with△(G)≥7.Moreover,it holds for any triangle free planar graph withΔ(G)≥6.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

On the hyperbolicity of the period-doubling fixed point

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.

     

颤振分析中的特征值排序问题

颤振分析中判断颤振临界速度的重要依据是系统V-g和V-f图,即系统特征值随参数的变化曲线.在几乎所有商用软件及自编程序的输出结果中,有时会出现所谓的"窜支"现象,这给颤振临界速度和颤振穿越分支及耦合形式的判断带来很大不便.通过隐函数定理可以证明,除重特征值点以外,系统特征值连续依赖于系统参数变化.依据多元向量值函数连续性,建立对特征值的排列算法,给出系统特征根轨迹的正确曲线,再输出V-g和V-f图数据,从而避免"窜支"现象.编制应用程序,通过几个典型算例对算法进行了验证.该工作能够有效简化颤振分析的后处理工作,提高分析效率.     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

树脂基复合材料粘接的优化设计

本文在试验数据和理论分析的基础上,提出树脂基复合材料单搭接的优化设计的原则,胶粘剂的选择、被粘物的设计,粘接长度的选择、胶层厚度的选择等.认为通过优化设计可以提高粘接强度,减轻产品的重量,提高其质量.     

EPR制度下的家电以旧换新回收策略优化模型

研究生产者责任延伸制度(EPR)约束下,家电生产销售企业集中决策下实施以旧换新进行废旧产品回收的问题。在对家电消费者市场结构进行分析基础上,建立以企业利润最大化为目标的实施以旧换新回收策略的优化模型,并根据我国家电行业EPR制度的特点,构建政府与企业间竞争的Stackelberg博弈模型,研究在政府环境效益最优时的补贴和基金征收标准下,企业以旧换新回收产品最优定价决策;同时,与企业不直接参与废旧产品回收时单一定价相应结果进行比较。结果表明:在EPR约束下,政府环境效益最优时的补贴和基金征收标准,对企业实施以旧换新回收策略的产品销量、总利润和废旧产品回收量的影响取决于企业成本结构、消费者结构、消费者效用和废旧产品回收处理能力。企业决策者需根据不同情形决定是否参与以旧换新回收实践。     

拍卖机制设计在区块链中的应用与挑战

区块链是新一代信息技术的重要组成部分,是分布式网络、加密技术、智能合约等多种技术集成的新型数据库软件。过去的十多年,区块链技术在全球范围内产生广泛影响。如今的区块链技术,已从最初的关注于解决货币和支付的去中心化问题,转入到解决市场的去中心化问题。智能合约的出现使得基于区块链技术的去中心化金融进入高速发展状态,也涌现出区块链环境下的各类拍卖场景。本文首次从机制设计角度,以区块链交易费机制,非同质化代币(Non-Fungible Token,NFT)拍卖和矿工可提取价值(Miner-Extractable Value,MEV)交易位置拍卖为主要对象,总结和剖析近些年来区块链上特有的拍卖机制;并针对区块链特性,提出区块链上拍卖机制设计所面临的挑战和未来亟待解决的问题。     

Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.     

Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.